## 金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|Market portfolio

Recall that the market portfolio is the optimal portfolio on the efficient frontier taking into account the existence of a risk-free asset. The line connecting the market portfolio with the risk-free asset is tangent to the minimum variance line and has maximal slope among the lines determined by all portfolios (see Figure 4.8).

In Chapter 2 we found the formula for the market portfolio obtained in the case of two risky securities determining the efficient set. This result is of course applicable to the general situation in view of Corollary $4.8$.

However, we derive the formula again; this time the parameters of all $n$ securities will be used.
Theorem $4.10$
If the risk-free return $R$ is smaller than the expected return of the minimum variance portfolio, then the market portfolio exists and is given by
$$\mathbf{m}=\frac{C^{-1}(\mu-R \mathbf{1})}{\mathbf{1}^{\mathrm{T}} C^{-1}(\boldsymbol{\mu}-R \mathbf{1})} .$$
Proof From Theorem $4.9$ we know that the minimum variance line is a hyperbola. Since its centre is on the vertical axis, there exists a single tangency point for a half line emanating from $(0, R)$, which maximises the slope (see Figure 4.8). The slope in question is of the form
$$\frac{\mu_{\mathrm{w}}-R}{\sigma_{\mathbf{w}}}=\frac{\mathbf{w}^{\mathrm{T}} \boldsymbol{\mu}-R}{\sqrt{\mathbf{w}^{\mathrm{T}} C \mathbf{w}}},$$
where $\mathbf{w}$ are the weights of a portfolio and $R$ is the risk-free rate of return. At the maximal slope the Lagrangian
$$L(\mathbf{w})=\nabla\left(\frac{\mathbf{w}^{\mathrm{T}} \boldsymbol{\mu}-R}{\sqrt{\mathbf{w}^{\mathrm{T}} C \mathbf{w}}}\right)-\lambda \nabla\left(\mathbf{w}^{\mathrm{T}} \mathbf{1}-1\right),$$
needs to be equal to zero. We can compute the gradients using Lemma $4.3$ and equate them to zero:
$$L(\mathbf{w})=\frac{\boldsymbol{\mu} \sqrt{\mathbf{w}^{\mathrm{T}} C \mathbf{w}}-\left(\mathbf{w}^{\mathrm{T}} \boldsymbol{\mu}-R\right) \frac{1}{2 \sqrt{w^{\mathrm{T}} C \mathbf{w}}} 2 C \mathbf{w}}{\mathbf{w}^{\mathrm{T}} C \mathbf{w}}-\lambda \mathbf{1}=\mathbf{0} .$$

## 金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|The Capital Asset Pricing Model

The market portfolio exists when the return on the minimum variance portfolio exceeds the risk-free return. The Capital Asset Pricing Model (CAPM) provides a linear relationship between the expected return $\mu_{\mathrm{m}}$ on the market portfolio and that of any risky asset. The two are linked by means of a parameter, commonly known as the beta $(\beta)$, providing a measure of undiversifiable risk of an asset. In the chapter we explore this relationship and show how the CAPM formula can assist investment decisions and introduce measures of portfolio performance.

Paradoxically, although we use variance to quantify risk, in assessing portfolio risk the variances of the assets in the portfolio turn out to be less relevant than their mutual covariances. To demonstrate this, let us consider the following example.
Example $5.1$
Suppose that the weights of a portfolio are of the form $w_j=\frac{1}{n}, j \leq n$, where $n$ is the number of assets in the portfolio. We investigate the risk of this portfolio in terms of its dependence on $n$. Assume that the variances of all securities on the market are uniformly bounded, $\sigma_j^2 \leq L$. Then
$$\sigma_{\mathbf{w}}^2=\sum_{j, k=1}^n w_j w_k \sigma_{j k}=\sum_{j=1}^n w_j^2 \sigma_j^2+\sum_{j \neq k} w_j w_k \sigma_{j k} \leq n \frac{1}{n^2} L+\frac{1}{n^2} \sum_{j \neq k} \sigma_{j k} .$$

Assume further that the off-diagonal elements of the covariance matrix are uniformly bounded, $\left|\sigma_{j k}\right| \leq c$, for some $c>0$. Then
$$\sigma_w^2 \leq \frac{L}{n}+\frac{1}{n^2} n(n-1) c .$$
The upper bound converges to $c$ as $n \rightarrow \infty$. Hence the risk of a portfolio containing many assets is determined by the covariances. The variances of the ingredients become irrelevant for large $n$.

This example motivates the following distinction between two kinds of risk: diversifiable, or specific risk, which can be reduced to zero by expanding the portfolio, and undiversifiable, systematic, or market risk, which cannot be avoided because the securities are linked to the market
From the above example we see that the variances of returns on individual securities are not the leading factors in determining the risk of a portfolio. The risk should rather depend on its undiversifiable risk, which should in turn depend on the asset’s covariances with the remaining assets. The aim of the Capital Asset Pricing Model (CAPM) is to quantify the systematic risk of an asset and to link it with its expected return.

# 风险和利率理论代考

## 金融代写|风险和利率理论代写市场风险、措施和投资组合理论代考|市场投资组合

$$\mathbf{m}=\frac{C^{-1}(\mu-R \mathbf{1})}{\mathbf{1}^{\mathrm{T}} C^{-1}(\boldsymbol{\mu}-R \mathbf{1})} .$$

$$\frac{\mu_{\mathrm{w}}-R}{\sigma_{\mathbf{w}}}=\frac{\mathbf{w}^{\mathrm{T}} \boldsymbol{\mu}-R}{\sqrt{\mathbf{w}^{\mathrm{T}} C \mathbf{w}}},$$
，其中$\mathbf{w}$是投资组合的权重，$R$是无风险收益率。在最大斜率处，拉格朗日量
$$L(\mathbf{w})=\nabla\left(\frac{\mathbf{w}^{\mathrm{T}} \boldsymbol{\mu}-R}{\sqrt{\mathbf{w}^{\mathrm{T}} C \mathbf{w}}}\right)-\lambda \nabla\left(\mathbf{w}^{\mathrm{T}} \mathbf{1}-1\right),$$

$$L(\mathbf{w})=\frac{\boldsymbol{\mu} \sqrt{\mathbf{w}^{\mathrm{T}} C \mathbf{w}}-\left(\mathbf{w}^{\mathrm{T}} \boldsymbol{\mu}-R\right) \frac{1}{2 \sqrt{w^{\mathrm{T}} C \mathbf{w}}} 2 C \mathbf{w}}{\mathbf{w}^{\mathrm{T}} C \mathbf{w}}-\lambda \mathbf{1}=\mathbf{0} .$$

## 金融代写|风险和利率理论代写市场风险、措施和投资组合理论代考|资本资产定价模型

$$\sigma_{\mathbf{w}}^2=\sum_{j, k=1}^n w_j w_k \sigma_{j k}=\sum_{j=1}^n w_j^2 \sigma_j^2+\sum_{j \neq k} w_j w_k \sigma_{j k} \leq n \frac{1}{n^2} L+\frac{1}{n^2} \sum_{j \neq k} \sigma_{j k} .$$

$$\sigma_w^2 \leq \frac{L}{n}+\frac{1}{n^2} n(n-1) c .$$

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